Constructing a polynomial whose nodal set is the three-twist knot 5_2

Mark Dennis, Benjamin Bode

Research output: Contribution to journalArticlepeer-review

8 Citations (Scopus)


We describe a procedure that creates an explicit complex-valued polynomial function of three-dimensional space, whose nodal lines are the three-twist knot 52. The construction generalizes a similar approach for lemniscate knots: a braid representation is engineered from finite Fourier series and then considered as the nodal set of a certain complex polynomial which depends on an additional parameter. For sufficiently small values of this parameter, the nodal lines form the three-twist knot. Further mathematical properties of this map are explored, including the relationship of the phase critical points with the Morse–Novikov number, which is nonzero as this knot is not fibred. We also find analogous functions for other simple knots and links. The particular function we find, and the general procedure, should be useful for designing knotted fields of particular knot types in various physical systems.
Original languageEnglish
Article number265204
JournalJournal of Physics A: Mathematical and Theoretical
Issue number26
Publication statusPublished - 6 Jun 2017

Bibliographical note

arXiv: 1612.06801


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